Introduction to Fluid Flow

Learning Outcomes

After this lecture you should be able to…Explain viscosity and how it changes with temperatureWrite the continuity equationDefine laminar and turbulent flow by using the Reynolds numberDetermine if a flowrate is laminar or turbulentWrite and Explain the Bernoulli equationApply the Bernoulli equation

Basics of Fluid Flow

A fluid is a substance that flowsWhen subjected to a shearing stress layers of the fluid slide relative to each otherBoth gases and liquids are defined as fluidsFluid mechanics is the study of the flow of gases and liquidsThe degree of resistance to shear stress is represented by the term ‘viscosity’High viscosity means high resistance to shear stress – does not flow easily

ViscosityDynamic Viscosity or Viscosity is a measure of resistance to shearing motionThe unit is Ns/m2…….but it has no name!The poise or centipoise is the SI cgs unit1 centipoise = 1 x 10-3 Ns/m2

Typical values for viscosityWater at 20°C = 1 cPAir at 20°C = 1.8 x 10-2 cPCrude Oil = 7.2 cPPetrol = 0.29 cP

You may hear the term ‘kinematic viscosity’This is dynamic viscosity divided by fluid densityIts SI cgs unit is the Stoke (= 1 cm2/s)NB – Viscosity is a function of temperature. For liquids, viscosity decreases as temperature increases

Basics Equations for Fluid Flow

The continuity equation Q = v.awhere v is the velocity (m/s) and a the area available for flow (m2 e.g. cross sectional area of a pipe) and Q is the flowrate (m3/s)The Reynolds number is used to define laminar and turbulent flowLaminar flow is defined by slow moving, uniform, even, smooth flow (e.g. a canal)Turbulent flow is uneven and rough (e.g. a white water river)Bernoulli equation. Daniel Bernoulli lived in the 18th

century and derived a relationship between velocity, height and pressure

The Continuity equation

Q=vaQ – flowrate, m3/sv – fluid velocity, m/sa – area available for flow, m2

What is the flowrate from your kitchen tap?(What is the volume of your kettle and how long does it take to fill it?)The pipe feeding the tap is 15mm. What is the cross sectional area?Use the continuity equation to determine the velocity

Continuity Equation contd.

Imagine a long pipe of varying diameter.The flowrate is constantWhere the diameter is large, the velocity is smallWhere the diameter is small, the velocity is large

d1v1

1 2

d2v2

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Osborne Reynolds 1842 - 1912

A pioneer in Fluid MechanicsHe discovered the nature of flow depends on

VelocityFluid physical propertiesGeometry of the channel/pipe

Sometimes flow is even and smoothSometimes it is uneven and roughHe asked Why?

Reynolds Experiment

He investigated fluid flow using this apparatus

Dye

Reynolds Experiment - Velocity

His first discovery ……At very low water flowrates, dye did not break upImplies no mixing between dye and water!

Dye

Reynolds Experiment - Velocity

….. And at high water flowrates, dye did break upDye mixed with water

Dye

Reynolds Concluded that

At low flowrates we get streamline or laminar flowFlow is characterised by streams that don’t mixAt high flowrates we get turbulent flow and a lot of mixing

Increase Velocity

Further Experiments - Viscosity

Reynolds heated the waterWhen heated the change from laminar to turbulent occurred sooner (at a lower velocity)This is explained by viscosityViscosity decreases as temperature increases

Decrease Viscosity

Further Experiments - Density

Reynolds replaced water with liquids of different densityThe change from laminar to turbulent occurred sooner for high density liquids

Increase Density

Further Experiments – Tube diameter

Reynolds used tubes of different diameterHe discovered that as the diameter increased the change to turbulent occurred sooner

Increase Diameter

Reynolds Number

He combined these observations into a dimensionless number which now carries his name

µρvd

=Re

Re = Reynolds numberρ = density (kg/m3)v = velocity (m/s)d = pipe diameter (m)µ = viscosity (kg/ms)

Activity – Laminar or Turbulent?

Is the flow from your kitchen tap laminar or turbulent?Determine the Reynolds No. and then use the table below

0 < Re <2000 Laminar flow 2000 < Re < 4000 Transition regionRe > 4000 Turbulent flow

Daniel Bernoulli (1700 – 1782)

Bernoulli was a pioneer in Science. His interests were medicine and engineeringBernoulli, with Leonard Euler, investigated the relationship between pressure and velocityThey punctured a pipe with a straw and observed that the height of liquid in the straw is related to the pressure in the pipeThis was used to measure blood pressure where patients arms were punctured with glass capillaries

Conservation of Energy

Bernoulli reasoned that the sum of pressure and kinetic energy is the same for any two points in a pipe

CPv =+2

21

ρ

This implies that if the velocity increases, pressure decreases.This is true for a horizontal pipe only.

Bernoulli Equation

Include a term for gravity, ρgh, to get the Bernoulli

Equation as follows

CghPv =++ ρρ 2

21

This is often written as follows:2222

2111 2

121

vghPvghP ρρρρ ++=++

Points 1 and 2 could be at two places in a pipe:

d1v1P1

1 2

d2v2P2

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Activity – Bernoulli Eqn Units

Determine the units of each term in the Bernoulli equation

CghPv =++ ρρ 2

21

Bernoulli Eqn Rearranged

Instead of expressing each term in units of Pressure, rearrange to give units of height

Chg

Pg

v=++

ρ2

2

How a chimney works

Point 1 is at the top of the chimney where the velocity is the same as the wind speedPoint 2 is in the fireplace where the velocity is almost zero

Activity – Flow in a pipe

A water mains supply enters a house at ground level (point 1) and rises vertically to the attic tank at an elevation of 10 m (point 2). No change in diamter.What is the ∆P?

10m

Point 2V = 2 m/s

Point 1

Activity – Bernoulli Eqn 2

Same as before except the pipe changes from 40mm diameter to 20mm. What is the ∆P?

10m

Point 220mmV = 2 m/s

Point 140mmV = ? m/s

Litre/s Litre/min m3/hr m3/s Ft3/hr Ft3/min gpm

1 Litre/s 1 60 3.6 0.001 127.1 2.119 15.85

1 litre/min 0.0167 1 0.06 1.66x10-5 2.12 0.035 0.264

1 m3/hr 0.278 16.67 1 0.00028 35.3 0.588 4.438

1 m3/s 1,000 60000 3,600 1 127,133 2,119 15,850

1 Ft3/hr 0.0078 0.472 0.0283 7.87x10-6 1 0.0167 0.124

1 Ft3/min 0.472 28.3 1.699 0.00047 60 1 7.481

Conversion Table